On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces

• Autorzy: Abass Hammed A. , Oyewole Olawale K. , Mebawondu Akindele A. , Aremu Kazeem O. , Narain Ojen K.

Identyfikatory

DOI

10.1515/dema-2022-0158

• Języki publikacji: EN

Abstrakty

EN

In this article, motivated by the works of Ali Akbar and Elahe Shahrosvand [Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932], Eskandani et al. [A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132], B. Ali and M. H. Harbau [Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces (2016) Article ID 5161682, 18 pages], and some other related results in the literature, we introduce a hybrid extragradient iterative algorithm that employs a Bregman distance approach for approximating a split feasibility problem for a finite family of equilibrium problems involving pseudomonotone bifunctions and fixed point problems for a finite family of Bregman quasi-asymptotically nonexpansive mappings using the concept of Bregman K-mapping in reflexive Banach spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution to the aforementioned problems. Furthermore, we give an application of our main result to variational inequalities and also report a numerical example to illustrate the convergence of our method. The result presented in this article extends and complements many related results in the literature.

Słowa kluczowe

EN

equilibrium problem; Bregman quasi-nonexpansive; pseudo-monotone operators; iterative schemes; fixed point problem

PL

zagadnienie równowagi; operator pseudo-monotoniczny; schemat iteracyjny; problem punktu stałego

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 658--675
• Opis fizyczny: Bibliogr. 34 poz., wykr.

Twórcy

autor

Abass Hammed A.

• School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
• DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

autor

Oyewole Olawale K.

• Department of Mathematics, The Technion - Israel Institute of Technology, 32000 Haifa, Israel

autor

Mebawondu Akindele A.

• Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria
• School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
• DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

autor

Aremu Kazeem O.

• Department of Mathematics, Usmanu Danfodiyo University Sokoto, P.M.B. 2346, Sokoto State, Nigeria
• Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, P.O. Box 60, 0204, South Africa

autor

Narain Ojen K.

• School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

Bibliografia

• [1] H. H. Bauschke and J. M. Borwein, Legendre functions and method of random Bregman functions, J. Convex Anal. 4 (1997), 27–67.
• [2] H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Essentially smoothness, essentially strict convexity and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), 615–647.
• [3] J. M. Borwein and Q. J. Zhu, Techniques of variational analysis, Canadian Math. Soc. 2005, 1–368.
• [4] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co. Inc., River Edge NJ, 2002.
• [5] L. M. Bregman, The relaxation method for finding the common point of convex sets and its application to solution of problems in convex programming, U.S.S.R Comput. Math. Phys. 7 (1967), 200–217.
• [6] B. Ali and M. H. Harbau, Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces 2016 (2016), Article ID 5161682, 18 pages.
• [7] H. A. Abass, F. U. Ogbuisi, and O. T. Mewomo, Common solution of split equilibrium problem with no prior knowledge of operator norm, U. P. B Sci. Bull., Series A, 80 (2018), no. 1, 175–190.
• [8] H. A. Abass, C. C. Okeke, and O. T. Mewomo, On split equality mixed equilibrium and fixed point problems of generalized ki -strictly pseudo-contractive multivalued mappings, Dyn. Contin. Discrete Impuls. Syst. B Appl. Algorithms 25 (2018), no. 6, 369–395.
• [9] E. Blum and W. Oettli, From Optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123–145.
• [10] D. V. Thong, P. Cholamjiak, M. T. Rassias, and Y. J. Cho, Strong convergence of inertial subgradient extragradient algorithm for solving pseudomonotone equilibrium problems, Optimization 16 (2022), 545–573.
• [11] G. Z. Eskandani, M. Raeisi, and T. M. Rassias, A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132.
• [12] P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear. Convex. Anal. 6 (2005), 117–136.
• [13] G. Kassay, M. Miholca, and N. T. Vinh, Vector quasi-equilibrium problems for the sum of two multivalued mappings, J. Optim. Theory Appl. 169 (2016), 424–442.
• [14] L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. 18 (1992), 1159–1166.
• [15] C. Bryne, Iterative oblique projection onto convex subsets and the split feasibility problems, Inverse Probl. 18 (2002), 441–453.
• [16] Y. Censor and T. Elfving, A multiprojection algorithms using Bregman projections in a product space, Numer. Algor. 8 (1994), 221–239.
• [17] P. Cholamjiak and P. Sunthrayuth, A halpern-type iteration for solving the split feasibility problem and fixed point problem of Bregman relatively nonexpansive semigroup in Banach Spaces, Filomat 32 (2018), no. 9, 3211–3227.
• [18] K. R. Kazmi, R. Ali, and S. Yousuf, Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces, J. Fixed Point Theory Appl. 20 (2018), 151.
• [19] O. K. Oyewole, H. A. Abass, and O. T. Mewomo, A strong convergence algorithm for a fixed point constrained split null point problem, Rendiconti del Circolo Matematico di Palermo Series 2 70 (2020), no. 1, 389–408.
• [20] F. Schopfer, T. Schuster, and A. K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Probl. 24 (2008), no. 5, 055008.
• [21] Y. Shehu, F. U. Ogbuisi, and O. S. Iyiola, Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces, Optimization 65 (2016), 299–323.
• [22] A. Akbar and E. Shahrosvand, Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932.
• [23] S. Timnak, E. Naraghirad, and N. Hussain, Strong convergence of Halpern iteration for products of finitely many resolvents of maximal monotone operators in Banach spaces, Filomat 31 (2017), no. 15, 4673–4693.
• [24] D. Butnairu and E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstract Appl. Anal. 2006 (2006), 84919, 1–39.
• [25] D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, 2000.
• [26] S. Reich and S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 24–44.
• [27] S. Reich and S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), 471–485.
• [28] J. V. Tie, Convex Analysis: An Introductory Text, Wiley, New York 1984.
• [29] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990.
• [30] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2000.
• [31] G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Èkon. Mat. Metody. 12, (1976), 747–756.
• [32] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006), 191–201.
• [33] M. A. Noor, K. I. Noor, and M. T. Rassias, New Trends in General Variational Inequalities, Acta Applicandae Math. 170 (2020), no. 1, 981–1064.
• [34] D. Q. Tran, M. L. Dung, and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization 57 (2008), 749–776.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).